The **Bachelier** **model** is the name given by a model of an asset price under brownian motion presented by Louis Bachelier on his PhD thesis *The Theory of Speculation* (*Théorie de la spéculation*, published 1900). It is also called “Normal Model” equivalently (as opposed to “Log-Normal Model” or “Black-Scholes Model”).

On the day of 2020-04-08, CME Group posted the note *CME Clearing Plan to Address the Potential of a Negative Underlying in Certain Energy Options Contracts*,^{[1]} saying that after a threshold on price, it would change energy options model from Geometric Brownian Motion model and Black–Scholes model to Bachelier model. In the day 2020-04-20, oil prices reached for first time in history negative values,^{[2]} where Bachelier model took an important role in option pricing and risk management.

The European analytic formula for this model based on risk neutral argument is derived in *Analytic Formula for the European Normal Black Scholes Formula* (*Kazuhiro Iwasawa*, New York University, December 2nd, 2001). ^{[3]}

## References

**^**“CME Clearing Plan to Address the Potential of a Negative Underlying in Certain Energy Options Contracts”.*www.cmegroup.com*. Retrieved 2020-04-21.**^**“An oil futures contract expiring Tuesday went negative in bizarre move showing a demand collapse”.*CNBC*. 15 December 2003. Retrieved 21 April 2020.**^**“Analytic Formula for the European Normal Black Scholes Formula”.*New York University*. 2 December 2001.